Question

What is the value of log 27^9?

Answer 1

Answer:

$\log_{27}9=\frac{2}{3}$

Step-by-step explanation:

Given : Expression $\log_{27}9$

To find : What is the value of the expression ?

Solution :

Let $y=\log_{27}9$

Apply logarithmic property, $y=\log_a x\Rightarrow x=a^y$

Here, a=27 and x=9

So, $9=27^y$

We re-write it as,

$3^2=(3^3)^y$

$3^2=3^{3y}$

Compare the base,

$3y=2$

$y=\frac{2}{3}$

Therefore, $\log_{27}9=\frac{2}{3}$

Answer 2

Answer:   The answer is $\dfrac{2}{3}.$

Step-by-step explanation: We are given to find the value of $\log_{27}9.$

Here, we will be using the following properties of logarithm.

$(i)~\log a^b=b\log a,\\\\(ii)~\log_ba=\dfrac{\log a}{\log b}.$

So, we will be solving the given problem using these two as follows.

$\log_{27}9=\dfrac{\log9}{\log27}=\dfrac{\log 3^2}{\log 3^3}=\dfrac{2\log 3}{3\log 3}=\dfrac{2}{3}.$

Thus, the required value is $\dfrac{2}{3}.$